The Hermitian Two Matrix Model with an Even Quartic Potential

We consider the two matrix model with an even quartic potential W (y) = y/4+αy/2 and an even polynomial potential V (x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M1. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4 × 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M1. Our results generalize earlier results for the case α = 0, where the external field on the third measure was not present. Received by the editor October 20, 2010. Article electronically published on September 20, 2011; S 0065-9266(2011)00639-8. 2010 Mathematics Subject Classification. Primary 30E25, 60B20; Secondary 15B52, 30F10, 31A05, 42C05, 82B26.